The masterpieces of John Forbes Nash Jr

Embedding the Heisenberg group into a bounded dimensional Euclidean space with optimal distortion

“…This creates an error term in (1.10), but it turns out that this error term is quite small and smooth (being a "high-high paraproduct" of ∇φ and ∇ψ, it ends up being far more regular than either φ or ψ, even with the presence of the derivatives) and can be iterated away provided that the initial frequency cutoff N 0 is large and the function ψ has a fairly high (but finite) amount of regularity (we will eventually use the Hölder space C 20,α to measure this). 7 This symmetry exploiting trick however comes with a cost: we were unable to use this scheme to also impose the orthogonality conditions Xφ • Y ψ = 0 and Xψ • Y φ = 0, which would otherwise have been quite useful in ensuring that the function ψ retains the required freeness and immersion properties upon iteration; this is because each of these equations fails to be symmetric on φ and ψ. Instead, we will have to perform a delicate analysis of how the wedge product Xψ ∧Y ψ evolves as one replaces ψ with ψ + φ, relying in particular on a careful computation of components of a certain pseudoinverse matrix.…”